The Congruence Subgroup problem

Lie groups and algebraic groups are important in many major areas of mathematics and mathematical physics. We find them in diverse roles, notably as groups of automorphisms of geometric structures, as symmetries of differential systems, or as basic tools in the theory of automorphic forms. The author looks at their development, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. Starting from Lie's theory of local analytic transformation groups and early work on Lie algebras, he follows the process of globalization in its two main frameworks: differential geometry and topology on one hand, algebraic geometry on the other. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter. The essays in the first part of the book survey various proofs of the full reducibility of linear representations of \\mathbf{SL}_2{(\\mathbb{C})}, the contributions of H. Weyl to representations and invariant theory for semisimple Lie groups, and conclude with a chapter on E. Cartan's theory of symmetric spaces and Lie groups in the large. The second part of the book first outlines various contributions to linear algebraic groups in the 19th century, due mainly to E. Study, E. Picard, and above all, L. Maurer. After being abandoned for nearly fifty years, the theory was revived by C. Chevalley and E. Kolchin, and then further developed by many others. This is the focus of Chapter VI. The book concludes with two chapters on the work of Chevalley on Lie groups and Lie algebras and of Kolchin on algebraic groups and the Galois theory of differential fields, which put their contributions to algebraic groups in a broader context. Professor Borel brings a unique perspective to this study. As an important developer of some of the modern elements of both the differential geometric and the algebraic geometric sides of the theory, he has a particularly deep understanding of the underlying mathematics. His lifelong involvement and his historical research in the subject area give him a special appreciation of the story of its development.

This book grew out of seminar held at the University of Paris 7 during the academic year 1985-86. The aim of the seminar was to give an exposition of the theory of the Metaplectic Representation (or Weil Representation) over a p-adic field. The book begins with the algebraic theory of symplectic and unitary spaces and a general presentation of metaplectic representations. It continues with exposes on the recent work of Kudla (Howe Conjecture and induction) and of Howe (proof of the conjecture in the unramified case, representations of low rank). These lecture notes contain several original results. The book assumes some background in geometry and arithmetic (symplectic forms, quadratic forms, reductive groups, etc.), and with the theory of reductive groups over a p-adic field. It is written for researchers in p-adic reductive groups, including number theorists with an interest in the role played by the Weil Representation and -series in the theory of automorphic forms.

The aim of the article is to show that there are many finite extensions of arithmetic groups which are not residually finite. Suppose G is a simple algebraic group over the rational numbers satisfying both strong approximation, and the congruence subgroup problem. We show that every arithmetic subgroup of G has finite extensions which are not residually finite. More precisely, we investigate the group H¯2(Z/n)=limΓ→H2(Γ,Z/n),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\bar{H}}^2(\\mathbb {Z}/n) = \\begin{array}{c} \\lim \\\\ {\\mathop {\\scriptstyle \\varGamma }\\limits ^{\\textstyle \\rightarrow }} \\end{array} H^2(\\varGamma ,\\mathbb {Z}/n), \\end{aligned}$$\\end{document}where varGamma runs through the arithmetic subgroups of G. Elements of {bar{H}}^2(mathbb {Z}/n) correspond to (equivalence classes of) central extensions of arithmetic groups by mathbb {Z}/n; non-zero elements of {bar{H}}^2(mathbb {Z}/n) correspond to extensions which are not residually finite. We prove that {bar{H}}^2(mathbb {Z}/n) contains infinitely many elements of order n, some of which are invariant for the action of the arithmetic completion {widehat{G(mathbb {Q})}} of G(mathbb {Q}). We also investigate which of these (equivalence classes of) extensions lift to characteristic zero, by determining the invariant elements in the group H¯2(Zl)=limt←H¯2(Z/lt).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\bar{H}}^2({\\mathbb {Z}_l}) = \\begin{array}{c} \\lim \\\\ {\\mathop {\\scriptstyle t}\\limits ^{\\textstyle \\leftarrow }} \\end{array} {\\bar{H}}^2(\\mathbb {Z}/l^t). \\end{aligned}$$\\end{document}We show that {bar{H}}^2({mathbb {Z}_l})^{widehat{G(mathbb {Q})}} is isomorphic to {mathbb {Z}_l}^c for some positive integer c. When G(mathbb {R}) has no simple components of complex type, we prove that c=b+m, where b is the number of simple components of G(mathbb {R}) and m is the dimension of the centre of a maximal compact subgroup of G(mathbb {R}). In all other cases, we prove upper and lower bounds on c; our lower bound (which we believe is the correct number) is b+m.

Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a “soft” proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg’s later proofs.

We extend to the case ofodd-dimensional spheres a theorem of Lubotzky, Phillips and Sarnak giving optimally equidistributed sets of points. The proof relies on the theory of automorphic forms and higher-dimensional Shimura varieties.

En théorie des groupes, le théorème de Kurosh est un résultat de structure concernant les sous-groupes d’un produit libre de groupes. Le théorème principal de cet article est un résultat analogue dans le cadre des relations d’équivalence boréliennes à classes dénombrables, que nous démontrons en développant une théorie de Bass-Serre dans ce cadre particulier.

This book is a companion to the author's previous book The books' ultimate objective is to prove algebraicity of the critical values of the zeta functions of automorphic forms on unitary and symplectic groups. In the course of the study of the zeta functions, many important results, which were obtained by the author during 1960-2000, are exposed.

Rankin-Selberg method in the theory of automorphic forms

Modular forms for the Weil representation of SL2 (ℤ) play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac-Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systems and to the Leech lattice.

According to Sakellaridis, many zeta integrals in the theory of automorphic forms can be produced or explained by appropriate choices of a Schwartz space of test functions on a spherical homogeneous space, which are in turn dictated by the geometry of affine spherical embeddings. We pursue this perspective by developing a local counterpart and try to explicate the functional equations. These constructions are also related to the L 2 -spectral decomposition of spherical homogeneous spaces in view of the Gelfand-Kostyuchenko method. To justify this viewpoint, we prove the convergence of p-adic local zeta integrals under certain premises, work out the case of prehomogeneous vector spaces and re-derive a large portion of Godement-Jacquet theory. Furthermore, we explain the doubling method and show that it fits into the paradigm of L-monoids developed by L. Lafforgue, B. C. Ngo et al., by reviewing the constructions of Braverman and Kazhdan (2002). In the global setting, we give certain speculations about global zeta integrals, Poisson formulas and their relation to period integrals.

For many applications of the trace formula to problems in the theory of automorphic forms one needs to stabilize the trace formula. Such a stabilization has been achieved by Arthur, building on work of Langlands and Kottwitz, subject to conjectures for orbital integrals known as fundamental lemmas. For the group Sp4 the standard invariant fundamental lemma has been established by Hales. However, for the stabilization of the full trace formula one needs to prove a fundamental lemma for weighted orbital integrals on Sp4. We prove this weighted fundamental lemma and thereby make Arthur's stabilization of the Sp4 trace formula unconditional.

The group theory, as one of the cornerstones of the modern algebra, has a profound historical trajectory that reflects the evolution of the mathematical thought. This comprehensive paper analyses the historical development of the group theory and provides an overview of the interconnectedness of the several key theorems in the group theory: The Lagranges Theorem, the Fermats Little Theorem and the Eulers Theorem. This paper begins by establishing the modern group-theoretical framework within the Lagranges Theorem on the link between the order of groups and that of its subgroups. Then, an extension onto other related theorems are provided. In all, this paper is highly interlinking among the ideas in group theory. Ultimately, this study not only demonstrates the beauty of mathematical interconnections but also highlights their continuing relevance for the modern applications, showing how the classical results remain relevant to guide contemporary explorations in algebra, number theory, and related disciplines.

Group theory is main topic modern algebra and group is also useful in many other fields. In this paper we show a special type relationship between group theory and number theory. In this paper will give type group of rational functions with coefficients as Fibonacci number with respect composition of mapping operation. We also show that is not only a group but also a cyclic group. In this represent a special relation between properties of group and rational functions with Fibonacci numbers coefficients. In this paper we gave a special type of recurrence relation sequence of rational functions with coefficients as Fibonacci numbers and also we proved the collection of all such rational function form a cyclic group with respect to the composition of function operation. In this represent a special relation between properties of group and rational functions with Fibonacci numbers coefficients. In this paper we gave a special type of recurrence relation sequence of rational functions with coefficients as Fibonacci numbers and also we proved the collection of all such rational function form a cyclic group with respect to the composition of function operation.

Mathematical ideas from number theory, group theory, dynamical systems, and computer science have often been used to explain card tricks. Conversely, playing cards have been often used to illustrate the mathematical concepts of probability distributions and group theory. In this paper, we describe how the 21-card trick may be used to illustrate concepts such as a discrete dynamical system, a fixed point, and the stability of a fixed point. The 21-card trick is a way of dealing cards in order to predict a card that is selected by a volunteer, within three moves. The 21-card trick and its two generalizations are examples of piece-wise linear, non-homogeneous, discrete dynamical systems, all of which have a global stable fixed point.

The general subject of the conference was geometric group theory, that is the study of groups via suitable actions on spaces endowed with additional structures, for instance a distance, a measure (class), or both of them with natural compatibilities. The interactions with other mathematical fields such as algebraic topology (e.g. via homological methods), functional analysis (e.g. via the theory of operator algebras), probability theory (e.g. via random walks) or even representation theory and number theory, play an increasing role in the development of this part of group theory.